Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $q = \dfrac{2k + 2}{-5k^2 - 25k - 20} \times \dfrac{k^2 + 19k + 90}{k + 10} $
Answer: First factor out any common factors. $q = \dfrac{2(k + 1)}{-5(k^2 + 5k + 4)} \times \dfrac{k^2 + 19k + 90}{k + 10} $ Then factor the quadratic expressions. $q = \dfrac {2(k + 1)} {-5(k + 1)(k + 4)} \times \dfrac {(k + 10)(k + 9)} {k + 10} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac {2(k + 1) \times (k + 10)(k + 9) } { -5(k + 1)(k + 4) \times (k + 10)} $ $q = \dfrac {2(k + 10)(k + 9)(k + 1)} {-5(k + 1)(k + 4)(k + 10)} $ Notice that $(k + 1)$ and $(k + 10)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac {2(k + 10)(k + 9)\cancel{(k + 1)}} {-5\cancel{(k + 1)}(k + 4)(k + 10)} $ We are dividing by $k + 1$ , so $k + 1 \neq 0$ Therefore, $k \neq -1$ $q = \dfrac {2\cancel{(k + 10)}(k + 9)\cancel{(k + 1)}} {-5\cancel{(k + 1)}(k + 4)\cancel{(k + 10)}} $ We are dividing by $k + 10$ , so $k + 10 \neq 0$ Therefore, $k \neq -10$ $q = \dfrac {2(k + 9)} {-5(k + 4)} $ $ q = \dfrac{-2(k + 9)}{5(k + 4)}; k \neq -1; k \neq -10 $